Here we deduce the reduction formula needed for integration of partial fractions with a power of a quadratic term. First we prepare the integral for integration by parts using the add-subtract trick:

The first integral we leave as it is, to the second we apply the by parts method:

By the way, when determining the function g we used the substitution z = y2 + 1.

When we substitute back, we obtain

And this is it. There is another way to deduce this formula: The integral is transformed using an indirect substitution (see the box "root from quadratics"):

The even power of a cosine is then handled using reduction formulas, but these are again deduced using integration by parts (see this note), so we in fact end up doing the same calculations as above, just in a different disguise.